Studies on surface hoar : formation and physical properties by Renee Maria Lang
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Engineering Mechanics
Montana State University
© Copyright by Renee Maria Lang (1985)
Abstract:
Field studies on surface hoar were conducted during the winter months of 1982-83 and 1983-84, at the
Big Sky Ski Area, Big Sky, Montana. Mechanical shear strength tests, conducted on established surface hoar layers, indicated that although a layer would become visually undetectable, shear strength remained too low to measure for extended periods of time.
The initiation of surface hoar growth was dependent on a variety of near-surface and atmospheric conditions. Nocturnal clear-sky radiative heat loss from the snow surface did not necessarily predispose condensation onto the surface, although near-surface air temperature gradients would be in excess of
+200°C/m.
A steady-state approximation for conservation of mass and momentum, in conjunction with the temperature data, predicts that surface crystal growth cannot be a diffusion limited process.

STU D IES UN SURFACE HOAR: F O R M A T IO N
A N D P H Y SIC A L PROPERTIES by
Renee Maria Lang
A thesis submitted in partial fulfillm ent o f the requirements for the degree of
Master o f Science in
Engineering Mechanics
M O N T A N A STATE U N IV E R S IT Y
Bozeman, Montana
May 1985

flrcU vveS
IU 3 1 S
U 5 L S 5
A PPR O VA L o f a thesis submitted by
Renee Maria Lang
This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citation, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies.
Dato
/ r / f / - r
^ / Chairperson, Graduate Committed
Approved for the Major Department
Head, Major Department ^ Date
Date
Approved for the College o f Graduate Studies y - "2 < - X T '
Graduate Dean

iii
S T A T E M E N T OF PERMISSION T O USE
In presenting this thesis in partial fulfillm ent of the requirements for a master's degree at Montana State University, I agree that the Library shall make it available to borrowers under rules o f the Library. Brief quotations from this thesis are allowable w ithout special permission, provided that accurate acknowledgment o f source is made.
Permission for extensive quotation from or reproduction o f this thesis may be granted by my major professor, or in his absence, by the Dean of Libraries when, in the opinion of either, the proposed use o f the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed w ithout my permission.
Signature
Date

iv
A C K N O W LED G M EN TS
I would like to express thanks to Brian R. Leo fo r his special photographic skills, and to Dr. Robert L. Brown fo r allowing me the freedom to investigate this topic.
The work reported here was done under National Science Foundation Grant No.
CEE 7318476.

V
TA B LE OF CONTENTS
Page
A P P R O V A L .......................................................................................................................................
S T A T E M E N T OF PERMISSION TO USE.................................................................................. iii ii
A C K N O W L E D G M E N T S ................................................................................................
TA B LE OF C O N T E N T S .............. .............................................................................................. v
L IS T O F F IG U R E S ...........................................................................................................................
A B S T R A C T .................................................................................................................................. viii vi
Chapter
ONE IN T R O D U C T IO N .....................................................................................................
TWO SITES A N D IN S T R U M E N T A T IO N ....................................................................
TH R E E CASE S T U D IE S ....................................................................................................... 11
I
6
Study 1,2 9 -3 0 January 1984 ............................................................................. 11
Study 2 ,2 7 -2 8 February 1984 ....................................................................... 16
Study 3 ,2 0 January—10 March 19 8 3 ............................................................... 23
FO UR A N A L Y S IS OF TH E C O N D E N S A TIO N PROBLEM........................................ 29
F IV E RESULTS A N D C O N C L U S IO N S ......................................................................... 41
L IT E R A T U R E C IT E D ............................................................................................................... 45 iv

vi
LIST OF FIG U R ES piHure
1. Crystal habit o f ice grown in the atmosphere based on temperature and excess vapour density (Kobayashi, 1 9 6 1 ).......................................................
Page
4
2. Group of surface hoar crystals exhibiting secondary growth features....... ...................................................................................................................
3. Thermocouple s ta c k ....................................................................................................
4. Use of shear frame........................................................................................................
5. Modified shear frame, underside v ie w ..................................................................... g
10
5
7
6 JgaJj3erature W'th respect t0 time at selected levels, 29-30 January
7. Accumulation of sector plates on the snow surface at approximately
1900 M ST, 28-29 J an u a ry ......................................................................................... 14
8 . Sector plates, 0300 MST, 28-29 January................................................................ 15
9. Time-averaged temperature profile for 2 0 0 0 -0 3 0 0 MST, 29-30
January...........................................................................................................................
10. Curve-fit temperature profile........................................................
17
-jg
11. Temperature with respect to tim e at selected levels, 27-28
February 1984 ............................................................................................................. 20
12. Time-averaged temperature profile for 2 0 0 0 -0 3 0 0 MST, 27-28
February....................................................................................................................... 21
13. Curve-fit temperature profile...................................................................................... 22
14. Upper portion o f a single surface hoar crystal "in situ" on the snow surface at 1 100 MST, 20 January 1983 ....................................................... 24
15. Same crystal as in Figure 14, after three hours of insolation.............................. 25
16. In situ surface hoar layer, 43 days o ld ..................................................................... 27
17. Single surface hoar crystal, 50 days o ld .................................................................... 28

vii
Figures
18. 5 (t) vs. t .........................................................................................................................
Page
32
19. Calculated near-surface thermal and concentration diffusion velocities for 29-30 J a n u a ry ..................................................................................... 38
20. Calculated near-surface thermal and concentration diffusion velocities, 27-28 February.......................................................................................... 39

viii
ABSTRA CT
Field studies on surface hoar were conducted during the winter months of 1982-83 and 1983-84, at the Big Sky Ski Area, Big Sky, Montana. Mechanical shear strength tests, conducted on established surface hoar layers, indicated that although a layer would become visually undetectable, shear strength remained too Io w to measure for extended periods of time.
The initiation of surface hoar growth was dependent on a variety o f near-surface and atmospheric conditions. Nocturnal clear-sky radiative heat loss from the snow surface did not necessarily predispose condensation onto the surface, although near-surface air temper­ ature gradients would be in excess o f + 200 ° C/m .
A steady-state approximation fo r conservation of mass and momentum, in conjunc­ tion w ith the temperature data, predicts that surface crystal growth cannot be a diffusion limited process.

I
CHAPTER ONE
IN T R O D U C T IO N
Surface hoar crystals are conventionally defined as plate-type ice crystals which form by deposition o f water vapour onto the snow surface during the night. Due to their lack of intercrystalline bonding and weak attachment to the snow surface, accumulations o f sur­ face hoar form layers which are mechanically weak in shear. When such layers are buried by a subsequent snowfall, they provide excellent failure planes. The disaggregate crystals may act as lubricating layers, increasing the potential for slab avalanches. These same type o f hoar crystals may also form a deposit on the surface of aircraft, changing the aerody­ namic properties of the wings sufficiently to cause take-off problems (Henson and Longley,
1944).
To my knowledge, no previous thorough quantitative study o f surface hoar growth has been conducted. Qualitatively, it has long been understood that surface hoar crystals normally develop during cold clear nights, when radiative cooling causes the snow surface temperature to fall below that of the contacting air. If the interfacial air becomes super­ saturated with respect to the contacting ice surfaces, condensation occurs.
Subsequent snowfall on a surface hoar layer results in a mechanically unstable condi­ tion in the snowpack, conducive to slab avalanches. It has been suggested that "in many cases, surface hoar instability is relatively short-lived, lasting fo r only one or tw o storms"
(Perla and M artinelli, 1976). This is a widely accepted hypothesis, yet relatively few clari­ fying measurements have been made.
The initiation o f an avalanche is inevitably linked to shear failure w ithin the snowpack. Hence, the shear strength o f the various layers of snow w ithin a seasonal snowpack

2 is a reliable indicator o f slab avalanche stability/instability, and has been investigated by many authors (Haefeli, 1939; Roch, 1966; Keeler and Weeks, 1967; Martinelli, 1971; V oitkovsky, 1977; Perla, 1977; Perla, Beck, and Cheng, 1982). Attempts to correlate shear strength to parameters such as snow density, temperature and crystal size have produced erratic results (e.g., Perla et al., 1982). The most significant comparative measure of shear strength within a given layer o f snow seems to be the degree o f intercrystalline bonding, which is directly related to crystal morphology.
Althoughall ice crystals which occur at atmospheric temperatures and pressure possess the same basic hexagonal crystalline structure (the hexad symmetry axis is the c-axis, and perpendicular to it are the three a-axes at 60° to each other), they exhibit a large variety o f shapes. The more spherical forms of ice crystals are sometimes referred to as equilibrium growth forms, since they are known to develop under isothermal or near-isothermal condi­ tions (Colbeck, 1982). These spherical grains are usually associated w ith a high degree of intercrystalline bonding. Within the snowpack, they develop at the expense o f the more faceted forms. Layers o f these grains or clusters will remain mechanically stable unless free water is present within the snowpack. Faceted forms o f ice crystals, or the kinetic growth forms, can be completely cohesionless. This is especially true in the case o f plate-type (i.e., preferential growth along the a-axes) ice crystals such as surface hoar. Furthermore, when the nucleation site of a surface hoar crystal is a pre-existing snow crystal, it is also ineffec­ tively bonded at that point. Hence, once incorporated into the snowpack, a layer o f sur­ face hoar crystals form a shear failure plane. The layer remains weak until the individual crystals metamorphose into a more mechanically stable form . This, of course, cannot occur unless the kinetic growth form becomes thermodynamically unstable with respect to its environment.
The basic crystal habit o f kinetic growth forms is governed by temperature (Nakaya,
1954; Shaw and Mason, 1955; Kobayashi, 1 9 5 7 ,1 9 6 1 ; Hallett and Mason, 1958). Observed

3 changes in habit are not a consequence o f changes in the diffusion field but are dependent on some temperature sensitive property o f the crystal surface (Mason e ta l., 1963). Under laboratory conditions, preferential a-axis growth was found to occur within tw o sharply defined temperature regimes (Fig. I ) , O0C to -4 ° C , and - IO 0C to -2 2 °C (Kobayashi, 1961).
The secondary growth features, such as the "feathery" or "dendritic" extensions o f plates, commonly associated with surface hoar (Fig. 2), are a consequence of degree o f excess vapour density. Hence, surface hoar growth should occur only at temperatures where the crystal may begin as a thin plate and at some degree o f supersaturation (Mason et al.,
1963).
However, the growth o f ice crystals under various laboratory conditions cannot be expected to describe the condensation process at a snow surface o f ever-changing tempera­ ture exposed to the atmosphere. The purpose of this investigation was to more fu lly under­ stand the formation, physical properties and subsequent metamorphism o f surface hoar.
To accomplish this, a detailed field study has been carried out. Shear strength tests were conducted on established surface hoar layers and observations were made in order to deter­ mine the conditions under which the layer would metamorphose to a more stable equilib­ rium form . A correlation of progressive surface hoar development on a snow surface, w ith near-surface snow and air temperatures, snow surface temperature, initial snow surface conditions, horizontal air motion, and general atmospheric conditions was determined. The temperature and condensation rate data were used concomitantly w ith expressions for conservation o f mass and momentum, and diffusive vapour flux in order to estimate some o f the broader aspects of mass transport phenomena at a snow surface.

4
Hollow prism
Thick plate
Solid column
Very thick plate c/a - » 0.8
H ollow prism
Solid column c/a - * 1 . 4
T em p e ra tu re (K )
Figure I . Crystal habit of ice grown in the atmosphere based on temperature and excess vapour density (Kobayashi, 1961).

5

6
CHAPTER TWO
SITES A N D IN S T R U M E N T A T IO N
Night-time observations o f surface crystal growth were made at a large, topographi­ cally flat area (elevation 2680 m) located within the Big Sky Ski Area, Big Sky, Montana.
The area is relatively free from any obstacles which could cause significant back-radiation to the snow surface. Measurements were made on numerous occasions during the winter months of 1982-83 and 1983-84, and in only a few cases did actual surface hoar develop­ ment occur.
Profiles of temperature were obtained hourly by using tw o separate stacks of copperconstantan thermocouples (Fig. 3 ). Temperatures were obtained at .5, 1.0, 1.5, 2.0, 3.0, and 4 .0 cm above the snow surface, at the surface, and at depths of 1.0 and 2.0 cm in the snow. During each experiment, one thermocouple was shielded with an aluminum cone in order to determine radiational effects on temperature measurements. Nosignificantchanges were detected at any level with this method.
Any significant near-surface horizontal air movement was detected by placing "flags" o f net-type material at various levels.
Time-lapse photography was used, w ith limited success, in order to obtain an estimate o f the condensation rate.
General weather patterns were also recorded.
Shear strength measurements on established surface hoar layers were conducted on an undisturbed north-facing slope located within the Big Sky Ski Area, Big Sky, Montana.
Due to the fragile nature o f surface hoar, it must be tested "in situ." Devices developed previously for testing the shear strength o f alpine snow have not displayed consistent

Thtrmocoup/99
I cm
D -------------
I cm
D fc>
D
.5 cm
.5 cm
.5 cm
.5 cm
I cm
I cm

8 results. Furthermore, they are not sensitive enough to give a measurable value o f shear strength for such low strength layers such as surface hoar. These mechanisms were there­ fore considered unsuitable. A modification of the device commonly referred to as the standard shear "fram e" (Roch, 1966) was used instead. This revised shear "fram e" (Brown and Oakberg, 1982) provides a more equally distributed force throughout the snow sample by introducing a series of small protrusions, each o f which absorbs a small portion of the compressive force exerted on the snow sample during testing (Figs. 4 and 5). The "p u ll" device and gauge are located at the base of the frame such that angular deviations from a direction parallel to the layer do not occur, thereby reducing "user dependence" and giving a more consistently accurate measure o f actual shear strength.
The area of the modified frame is .0001 m 2 . The maximum variation on a given layer at a certain stage, when performing ten tests as suggested by Perla (1977), was less than
200 Pa. A slow "rate of p u ll" was used and tests were not limited to one "user."
Shear strength measurements and a photographic record o f the deterioration of the hoar were made on a weekly basis.

9
S N O W P l T
WALL
# SHEAR F R A M E iXjX
L AYER TO BE T E ST ED
S NO WP IT WALL

CHAPTER TH R E E
CASE STU D IES
The results of three representative field studies are described. The first and second are nighttime growth studies in which surface hoar was deposited to an average height o f 1.3
to 1.5 cm. It should be mentioned that in over 60% of observed cases, surface crystal growth did not occur under nocturnal clear-sky conditions. In many cases, needle-type growth (i.e., preferential growth along the c-axis) occurred, but shortly after sunrise the needles would be sublimated away. This may serve to explain w hy surface hoar is always described as plates (see for example, LaChapeIle, 1969), while the term depth hoar is used to describe any kinetic growth form which develops within the snowpack. In all cases, measured temperature profiles were similar unless a strong breeze persisted or cirruform interferred. Time-averaged temperature profiles over the period o f estimated maximum condensation were curve-fit by use o f the Simplex method (Nedler and Mead, 1965; Caceci and Cacheris, 1984), adapted to run under the V A X /V M S operating system.
The third case study describes a series o f shear strength tests and a photographic record o f an established surface hoar layer.
Study 1 ,2 9 -3 0 January 1984
1500 MST — The snow surface was composed of freshly fallen, stellar and rimed-stellar snow. A heavy cumulus cloud cover, accompanying the passage o f a cold front had lingered all day.
1800 MST — Only a few cirrus and altocumulus Ienticularis cloud forms remained. The air between 1 .0 and 4 .0 cm above the surface was nearly isothermal at approximately

12
-1 3 .0 °C , although a gradient in excess of + 200°C/m had already been established between the snow surface (-14 .8°C ) and air .5 cm above the surface (-1 3 .4 °C ). A negative temperature gradient had been established in the snow adjacent to the surface; the snow temperature at 2.0 cm below the surface was approximately - 1 4 .1 0C
(see Fig. 6 for temperature transitions at selected levels with respect to tim e).
1900 MST — Skies were perfectly clear. The snow surface had cooled to -1 6 .6 ° C. The adjacent snow and air were cooling at slightly lower rates. No hoar growth was notice­ able, but macro-photography revealed that sector plates had begun to accumulate
(Fig. 7).
2000 MST — Skies remained clear. The snow surface reached its minimum tempera­ ture o f - I S T 6 C. The air-snow surface temperature gradient was nearly linear at approximately + 1 4 0 °C/m . Similarly, the snow-snow surface temperature gradient was nearly linear at approximately - 8 0 ° C/m . Sector plates had accumulated to an average height o f .25 cm.
2300 MST — Skies remained clear. The snow at the measured depth 1.0 cm had become nearly isothermal w ith respect to the snow surface. The maximum tempera­ ture difference between the snow surface (-18 .5°C ) and the air at +4.0 cm (-1 2 .8 °C ) had been reached. The hoar crystals were approximately .5 -.7 cm high.
0000 MST — Skies remained clear. The maximum positive temperature gradient
(+ 3 8 0°C/m ) between the original snow surface {-16.4°C ) and the adjacent air at +.5
cm (-1 4 .5 °C ) was attained. (Note: in most other case studies when condensation occurred the gradient at this tim e would be in excess of + 3 0 0 °C/m .) A positive gradi­ ent (+ 7 0 °C/m ) had been established between the snow at -2 .0 cm (-1 7 .8 ° C) and the snow surface (-1 6 .4 °C). (This phenomenon o f temperature gradient reversal was also consistently observed to occur.)

13
—14
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<D k -
3
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0 ) l -
CO
Ir —16
0 )
Q
-10
-1 2 x
O
x o
D
X
D
O
8 s
O
O e e
A
▲
- 1 8
* S
A A
A A
- 2 0
1800 2 1 0 0 0 0 0 0 0 3 0 0
I
060 0

14

15

16
0 30 0 MST — Skies remained clear. The hoar plates reached their maximum size o f
1.3 -1 .5 cm in height (Fig. 8 ). No secondary growth features were observed.
0 6 0 0 M ST — The experiment was terminated. No horizontal air motion within 1.0 m o f the snow surface had been detected all night. A positive temperature gradient in both the snow and air persisted until this time. Skies had remained clear all night.
A time-average of the overall temperatures during the maximum growth period
(2 0 0 0 -0 3 0 0 M ST) is represented in Figure 9. Note that the overall effect of the temperature gradient reversal in the upper 2.0
cm of the snow is a near isothermal time-average. A curve-fit of the time-averaged temperature is represented in Figure 10.
Study 2, 27-28 February 1984
1500 MST — The snow surface was composed o f stellar snow resulting from a storm which had occurred on 25-26 February. Cirrus cloud forms were present and a slight breeze persisted. Due to the frontal passage and the initial snow surface temperature
(-9 .7 ° C) hoar growth was expected. Two additional tests were conducted. A black plastic sheet I m J was placed onto a section of the snow surface, the plastic being intended to form a vapour barrier between the snow and the atmosphere. Hence if the vapour flux from the snow to the snow surface was significant in supplying vapour for condensation, then the amount of condensate onto the plastic should be minimal.
Also, a stainless steel substrate 20 cm 2 was placed above the snow surface in order to determine how effectively the condensate would bond to the metal surface. O f course, due to the diverse properties o f the various surfaces, the net accumulation of hoar was expected to be quite variable.
1800 MST - The breeze was persisting, but the skies had cleared. A temperature grad­ ient o f approximately + 3 80 °C/m had been established between the snow surface
(-15 .8°C ) and the air at .5 cm above the surface (-1 3 .9 °C ). A negative temperature

17
- 1 2 T
O
O
0
3
4 - "
CD
Q.
E
CD h -
"O
CD
O) cd k _
CD
> cd
1
CD
E
F
—1 4 - -
—1 6 •
( I
- 1 8 -
- 2 . 0 0 2 .0 4 .0
Position with respect to Snow Surface (cm)
Figure 9. Time-averaged temperature profile fo r 2 0 0 0 -0 3 0 0 M ST, 29-30 January. Zero position is original snow surface.

18
2 6 0 -
2 5 8 -
2 5 6
2 5 4 -
-
2 .0
Position with respect to Snow Surface (cm)
Figure 10. Curve-fit temperature profile. For z > 0 , 6 = (6 .86 z/(2 .64 + z)) + 255.55.

19 gradient (approximately - 9 9 ° C/m ) had been established in the snow adjacent to the surface.
1900 MST — No breeze was detectable. The snow surface had continued to cool to approximately -1 8 .7 ° C. The air had cooled very rapidly at the levels less than 2.0 cm, such that the temperature gradient in the air was approximately linear at + 161eC/m .
Similarly, a nearly linear negative gradient in the adjacent snow existed (approximately
-1 0 8 ° C /m ). (See Fig. 11 for temperature transitions w .r.t. tim e.) Skies were clear.
210 0 MST — Sector plates had accumulated to an approximate average height of 0.5
cm on all surfaces. Skies remained clear and a t this tim e a very large temperature d if­ ference existed between the original snow surface (-1 9 .6 ° C) and the air at 0.5
cm
(-16 .5°C ).
2200 MST — The air and snow at all levels reached their minimum temperatures. The snow surface temperature was at -2 0 .5 ° C. Skies were clear.
010 0 MST — The temperature gradient in the snow had reversed in the upper 1.0 cm to + 5 °C/m . The plates were approximately 1 .0 -1 .2 cm in height on all surfaces. Skies remained clear.
030 0 MST — The hoar plates reached their maximum average height o f 1.2 -1 .5 cm.
No secondary growth features were observed.
060 0 MST — The experiment was terminated. Skies had remained clear throughout the night. No horizontal air motion was detected after 1900 MST. The plates were approximately the same size on all surfaces. The plates which had condensed onto the stainless steel substrate were more uniform ly oriented in a surface normal direction, and were indeed firm ly attached at their nucleation site.
The time-averaged temperature profile and curve-fit profile during the period o f maximum observed condensation are presented in Figures 12 and 13, respectively. In

20
-1 0
- 1 2
- 1 4
O
A
O o
3 - 1 6
.CD
x -
<D a
o
I -
-1 8
A
O
• •
* 6
O
A A
A
*
- 2 0
- 2 2
1 8 0 0 2 1 0 0 OOOO 0 3 0 0
0 6 0 0

21
- 1 2 T
O
<D
3
* - •
CO
L-
0 )
Q.
E
0 )
Y -
T J
O
O)
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0
>
<
1
<D
E i-
— 1 4
— 1 6 -
-u
-
2 .0
2 . 0 4 . 0
Position with respect to Snow Surface (cm)
Figure 12. Time-averaged temperature profile for 2 0 0 0 -0 3 0 0 MST, 27-28 February.

22
2 6 0 -
2 5 8
2 5 6 -
2 5 4
-
2 .0
Position with respect to Snow Surface (cm)
Figure 13. Curve-fit temperature profile. For z > 0, 6 = (10.77 z /(3 .6 6 + z)) + 253.35.

23 this particular case, a net negative temperature gradient had persisted in the upper
2.0
cm of the snowpack.
Study 3, 20 January-10 March 1983
20 January — Extremely well-developed surface hoar, averaging 4 .0 -5 .0 cm in height, had been deposited onto the snow surface during tw o previous nights (18-19 and
19-20 January). Tim e lapse photography of a single crystal in situ, between 1 100 and
1400 MST, during which time clear-sky conditions persisted, was conducted in order to determine the effects o f insolation on the crystal. The photographs revealed that the typical "feathery" or dendritic secondary growth features o f the hoar are an aggre­ gation o f separately developing sector plates oriented either on or at 60° to the pre­ dominant a-axis (Fig. 14), as reported by Mason et al. (1963). Three hours o f insola­ tion served to reduce the individual peripheral plates by insignificant amounts in comparison to the total size o f entire crystal (Fig. 15). A temperature gradient, in excess o f +200° C/m , between the base and top o f the crystal, persisted throughout this time.
27 January — The layer was covered by 10 cm o f low density snow. Structural changes o f the surface hoar were minimal. It was very well adhered to the adjacent snow above, but shear strength at the base of the layer was too low to measure (i.e., less than 25 Pa).
3 February — The peripheral edges of the crystals had rounded and pore spaces had become loosely filled with granular snow. Basal shear strength had increased to an average o f 35 Pa.
11 February — The layer was buried under 46 cm o f low density snow, but was still recognizable, although individual crustals had deteriorated into a more "axial" form.
The average basal shear strength had slightly decreased to an unmeasurable average.

24

25

26
4 March - The layer had become nearly "hollow " (Fig. 16), i.e., the crystals had deteriorated such that only the largest remained visible. The snow below was now composed entirely of well-rounded grains. Basal shear strength had increased to approximately 1 10 Pa.
10 March — It was very difficult to detect the layer, yet the crystals were still quite large (Fig. 17). The layer was obscured due to the degree o f rounding o f the hoar edges and bonding o f the hoar to well-rounded grains which had completely filled the pore spaces. Average basal shear strength was 390 Pa.
A fter this date, free water in the snowpack destroyed the layer.

27

28

29
CHAPTER FOUR
A N A L Y S IS OF T H E C O N D E N S A TIO N PROBLEM
Due to the complexity of even the simplest of atmospheric flows, some preliminary assumptions must be made. As a first approximation, it will be assumed that motion occurs only in the surface-normal direction, taken to be the z-coordinate direction.
One main difficulty is the definition o f some precise physical boundary, i.e., the exact position of the "snow surface" at some tim e t. In order to make the problem determinate, it will be assumed that the initial snow surface is located at z =0 at some tim e t = 0 , prior to condensation. A t some tim e t > 0 , the snow/atmosphere interface is located at 6 ( t ) > 0 , i.e., the growth rate o f each single ice crystal is assumed to be uniform . Hence, the interface is defined as the small region where the phase change from vapour to ice (or vice versa) is occurring.
The mass flu x to /fro m the underlying snow at z < 0 is assumed to be negligible. Motion in the x and y directions is neglected.
Transport properties were extrapolated from experimentally obtained values (Mason and Monchick, 1963). Saturation state values were extrapolated from Keenan and Keyes
(1978).
The temperature profile above z = 0 was taken to be constant w ith tim e at the onset of condensation, and values were obtained from the curve-fit profiles (shown in Figs. 10 and
13).
In any non-isothermal multicomponent mixture of a = 1 , 2 , . .. n constituents, mass must be conserved for each constituent and also for the mixture. Hence n+1 equations may be written for conservation o f mass, o f which only n are independent.

30
Assuming that any vapour being removed at the snow surface is being replaced from above, conservation of mass for steady, one-dimensional flow o f vapour may be w ritten as
( I ) where pv is the local or dispersed density o f the vapour, zy is the velocity o f the vapour, and mv is defined as the mass "exchange" or "production" (see for example Luikov and
Mikhailov, 1965) for the vapour, and represents a transfer from vapour phase matter to solid phase, or vice versa.
Since the ice essentially retains no velocity, conservation of mass for the ice phase matter may be expressed as
" 5 T * * = where p$ represents the dispersed density of the ice and may also be expressed as
( 2 )
(3) where 7 j is the mass density of the ice grain itself and y. is the volume distribution func­ tion o f the solid phase matter.
Conservation of mass for the air is
(4) then for the entire mixture
(5)
Obviously, this requires m$ + mv = 0 , which in turn gives the following relationship;
(
6
)
(7)

31
Then, assuming that the tim e rate of change o f the dispersed ice density is constant for steady state conditions, pvi v = C1Z + C 2 ( 8 )
- p $ = C i t + C 2 (9) where C1, C2 , C 3 are constants. Define m as the condensation rate, a as the condensation coefficient or the fraction of the vapour flux impinging on the snow surface which actu­ ally condenses, and d 8 /d t as the tim e rate of change o f the position o f the interface. Then
-a p v zv = m , Z = 5 (
10
)
(
11
)
Combining ( 8 ), (9), (10), (11), and using the initial condition o f p = 0 at t = 0 for z > 0 .
(
) C1I — - OC1S = o C dt
2 , which is a linear first order differential equation with the solution
5 exp (/— dt) = / t C 11 exp ( / — dt) dt + C t
4 , (13)
C 4 a constant. Integration of the above and use of the initial condition 5 (0) = 0 and final condition 5 (tf ) = 8 f , the interface position at some tim e t may be expressed as
5 = Sz (— ) . tf
(14)
Results for S as a function of tim e for various values o f the condensation coefficient are given in Figure 15. The tim e span o f 7 hours is an estimate based on observation. The low value o f e = .015 is based on an experimentally obtained value o f a = 0.0 1 44 ± 0 .0020 for solid ice between -1 3 °C and - 2 eC (Delaney, Houston, and Eagleton, 1963), as opposed to values near unity reported by other authors (see Hobbs, 1974).

32
Figure 15. 6 (t) vs. t.
Time (hrs.)

33
Differentiation of (14) and combining with (11) gives an expression fo r the dispersed ice density as follows;
(15)
'f which implies by use of (9) that r h /t ff is constant.
In order to obtain an estimate on the diffusive flux of vapour, consideration must first be given to a momentum equation for the binary gas mixture above 5. It is assumed that the stress in the fluid obeys the Navier-Poisson law for a fluid w ith no bulk viscosity.
Furthermore, assuming that the only body force is gravity and neglecting local acceleration, a momentum equation may be written as
-d p
"dz"
. d2z . dz
4/3,1 dz* " pS 9 = V S I '
(16) where /i is the dynamic viscosity and z is the vertical component of the barycentric or con­ vective velocity of the gas mixture (see for example, Malvern, 19 69). Also
(17) Pa + Pt is the total gas density, and
P = Pa + Pv (18) is the total gas pressure. It is also allowable to express the total gas pressure p as a sum of the static and dynamic contributions, such that
P = Pd + Pst • (19)
Then dp dPd dz dz d Psi dz
(
20
) dp dz " 5 7 - " s 9
( 21 )

34
Substitution o f (21) into (16) gives
- dPd dz
(22) which gives the dynamic pressure gradient in terms of the inertial and viscous forces. How­ ever, a dimensional analysis o f ( 22 ) readily shows that the dynamic pressure gradient is at least five orders o f magnitude less than the static pressure gradient. Therefore the total pressure gradient may be approximated as the hydrostatic equation, dp
5 7 = -PgS (23)
A diffusive mass flux per unit tim e and area for the vapour, j v, is defined by iv = P v ( Z y - Z ) (24)
= Pvuv (25) where again z is the mixture velocity. Therefore uv is defined as the diffusion velocity of the vapour. Since
- - ( P v Z v + Pai a) (26) obviously
The diffusive mass flu x results from "mechanical and thermal driving forces" (Bird,
Stewart, and Lightfoot, 1960; de Groot and Mazur, 1962), and may include terms from ordinary (concentration) diffusion, pressure diffusion, forced diffusion and/or thermal diffusion. In this analysis, the primary "driving forces" are related to the concentration and temperature gradients. By assuming a linear relationship o f the diffusive flu x to the driving forces, an expression for the diffusive mass flux o f vapour may be written as d C V 0 V9
- O , 0, , "57 * 'g —
dz
(28)

where c is the concentration of vapour, i.e.
35
Ctf = pg
(29)
0 is the absolute temperature, Dva is the binary diffusion coefficient, and D y0 is the thermal diffusion coefficient (Grew and lbbs, 1952). The Onsager reciprocity relations give
^av “ ^va (30)
'v (31)
(de Groot and Mazur, 1962; Jost, 1960). The expression fo r j y may be rewritten by defin­ ing the thermal diffusion factor for the vapour as
(32)
(Bird, Stewart, and Lightfoot, 1960), where ca = 1 - cv •
Substitution of (32) and (33) into (28) gives
(33)
L dz
K t f Cy ( I - C v )
0 d e l
(34)
A t such low temperatures and pressures, it is allowable to approximate the behaviour o f both the air/vapour mixture and the vapour itself as ideal, i.e.,
P = PgR e (35) and
Pv = Pv Rv @ (36) where R and Ry are the gas constants for the mixture and vapour respectively. Then (29) becomes
P„ R
(37)

36
Differentiation of the above gives
dz
p dz P 2 dz
(38)
Use of (17) and (18), and substituting (2 3 ), (35) and (36) into the above gives
p » R» Q
pvg
) (39)
Collecting terms, and again using (2 9 ), an expression for the concentration gradient of vapour may be written as
dz 0 R
Rv
(40)
A t such low temperatures
I - cv = I .
(41)
Substitution o f (40) and (41) into (34), and dividing both sides by p v gives an approximate expression for the diffusion velocity of the vapour; uv = "Ova
(42)
It is allowable to then separate the effects due to concentration and thermal diffusion by letting the contribution to uy by ordinary diffusion be
- D v a 9 I I
0 • r r
: 1 and by thermal diffusion be
- D va * v d 0 u, 9 ---------- — "57 such that
•
(43)
(44)
(45)

37
Using the curve-fit temperature profiles in Figures 10 and 13, results for uvc and Uvfl with respect to position are shown in Figures 19 and 20.
Two separate results are quite obvious. First, thermal diffusion is quite dominant, such that uv = uvfl • (46)
Regardless, the diffusive mass flu x is totally insufficient to produce any significant amount o f condensate, even with incredulous supersaturations. Therefore, convection must be the dominant mechanism for vapour transport. It then follows that the vapour velocity may be approximated as the convective velocity, i.e., zv = z . (47)
Then by (24), (34) and (4 7 ),the concentration gradient o f vapour is related to the temper­ ature distribution as follows;
- I
e dz
(48)
Neglecting the temperature dependence of k
, integration o f the above gives
C 6 9
(49) where C 6 is a constant, and must be evaluated by assuming some condition o f supersatu­ ration. Substitution o f (10), (29), (47), and (49) into (15) gives an approximate expression for the dispersed ice density as
-Jia(S)C6 I e ( S ) ] ' s V t 1- 0 Z(S)
PsIt I a - i ------------------- -------- -------------------- . ,50 )
This expresses the dispersed ice density at some tim e t as a function o f the gas density, temperature and the vertical component o f the convective velocity at the interface, degree of supersaturation and values o f the thermal diffusion factor and condensation coefficient.

38
- I Xl O
- 1 X 1 0
- 1 X 1 0
- 1 X 1 0
- 1 X 1 0
Figure 19. Calculated near-surface thermal and concentration diffusion velocities fo r 29-
30 January.

39
- 1 X 1 0
- I X l O
- 1 X 1 0
- I X l O
- i X l O
Figure 20. Calculated near-surface thermal and concentration diffusion velocities, 27-28
February.

40
An order of magnitude estimate on z , gives z = - I X l O ' 2 m/s to support a sufficient amount o f condensate for surface hoar growth.

41
CHAPTER F I V E
RESULTS A N D CO NCLUSIONS
Observations in the condensation studies agreed very well with transition temperatures reported by Kobayashi (1961), which induce changes in preferential growth along the aaxesto c-axis growth, or vice versa. When the snow surface cooled to temperatures measured to be less than -2 1 °C the end result was needle-like growth on the snow surface. During plate-like growth, measured snow surface temperatures ranged approximately between
-1 2 .5 °C and -21 eC. On numerous occasions, plates would begin to form on the snow sur­ face, but as the snow surface temperature would continue to decrease, needle-type growth would predominate.
In either case, the crystals would retain the orientation o f their initial nucleation site; an axis o f an existing surface crystal. Sector plates or needles were usually oriented within a few degrees of surface normal, along the temperature gradient. Crystals exhibiting secon­ dary growth features, such as dendritic hoar or rimed needles were more randomly oriented with respect to the snow surface, but within 6 0° o f surface normal.
It was, indeed, surprising that the net accumulation o f ice crystals on a variety o f surfaces was macroscopically equivalent. So although initially some vapour is probably transported through the underlying snow to the snow surface, it seems to have a negligible effect on the net amount o f condensate.
Furthermore, a large near-surface temperature gradient, due to nocturnal clear sky conditions, is insufficient in itself for significant condensation onto the snow surface to occur. A ny detectable level o f horizontal air motion near the surface was observed to pro­ hibit growth. However, the total estimated diffusive flux o f vapour is physically insufficient

42 for the amount o f condensate, even assuming very high supersaturations. Realistically, the total vapour flux must be orders o f magnitude larger than the resulting diffusive flux, which suggests that the actual vapour transport process is dominated by convection result­ ing from local temperature inequalities. The process o f hoarfrost formation may there­ fore be a quasi-turbulent phenomenon. If the air is calm, the loss o f vapour to the con­ densate could not be replenished rapidly enough by diffusion alone and condensation would cease. On the other hand, if air motion was too vigorous, the radiative cooling o f the snow surface would be offset by turbulent warming and the vapour inversion would be destroyed. Furthermore, hoar growth would be prohibitive due to the fragility o f the crystal itself.
Cloud cover conditions during the day must also be taken into consideration. If over­ cast skies prevail during the day, then subsequent clearing of the cloud cover at night causes the snow surface to cool rapidly to the frostpoint. Such conditions normally occur after the passage o f a cold front, during which tim e the air adjacent to the snow surface acquires a high humidity due to heavy cloud cover and/or recent snow accumulation associated w ith the front. A fter the frontal passage, air pressure rises, skies clear, and temperature drops. (However, this sequence of events need not be endemic to more humid regions, or prerequisite during spring conditions, when surface hoar may develop at temper­ atures near O0C.)
It is also noteworthy to report that although the net loss of heat from various ground surfaces during a high cirrus-type cloud cover is reported to be as great as when the sky is clear (Henson and Longley, 1944), the presence of high cirruform did interfere w ith long­ wave radiational cooling o f the snow surface, and hence vapour flux to the surface. Simil­ arly, it was observed that even during rapid growth rates, hoar would not form within small concavities on the snow surface, which implies that back radiation from w ithin an incurva­ ture is sufficient to prevent adequate surface cooling.

43
Another point of interest was that when the condensate was in the form of needles, shortly after sunrise they would be rapidly sublimated in areas exposed to insolation. But plate-type crystals were nearly unaffected by insolation, as discussed in Study 3 . It has been suggested that the growth rate (and hence the sublimation rate) o f kinetic growth forms is governed by the surface kinetics, which is reflected in variations o f the conden­ sation coefficient with respect to temperature and the c- and a-planes of ice (Lamb &
Hobbs, 1971). Some exact information on the variation of condensation and sublimation coefficients w ith axial orientation and temperature could partially serve to explain the instability of needles vs. the stability of plates during insolation.
Once a surface hoar layer is incorporated into the snowpack, the assumption that the resulting instability rapidly subsides is extremely dangerous. Hoar layers may remain indef­ initely in progressively more deteriorated forms, obscured by the presence o f rounded grains which have either "sifted dow n" into the pore spaces or perhaps formed from the mass lost from the perimeters of the hoar crystals themselves. O f course, the smaller the hoar crystals, the more rapidly they are obscured. In some cases, hoar layers may be nearly unnoticeable, even at the snow surface immediately after their formation. Unfortunately, the initial basal shear strength o f all surface hoar, whether a small sector plate or a large dendrite, is nearly non-existent.
One critical missing piece of information is the temperature profile through the hoar layer after its incorporation into the snowpack. During the winter months, a net negative temperature gradient is established in the snowpack. The high degree of porosity within a surface hoar layer combined with a negative temperature gradient should precipitate the development of similar kinetic growth form , depth hoar, at the expense o f the surface hoar. Y et this was not observed to occur. Furthermore, observations indicate that marked deterioration o f a surface hoar layer is effected only when the snowpack becomes iso­ thermal at temperatures near O6C.

44
In summary, it is sufficient to say that more information on the various properties o f snow and ice is needed. Mass transfer processes at the snow surface and within the snowpack are not easily explainable and demand further attention. Steady one-dimensional flow is, at most, a poor description of the problem at hand.

45
L IT E R A T U R E C ITE D

46
L IT E R A T U R E C ITED
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Colbeck, S. C. (1982). Growth of Faceted Crystals in a Snow Cover, C R R E L Report 82-29.
de Groot, S. R., and Mazur, P. (1962). Non-Equilibrium Thermodynamics. North Holland
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Delaney, L. J., Houston, R. W., and Eagleton, L. C. (1964). The Rate o f Vaporization of
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Haefeli, R. (1939). Snow Mechanics w ith References to Soil Mechanics, Bader et al., Der
Schee und seine Metamorphose, Beitrage zur Geologie der Schweiz, Geotechnische
Serie, Hydrologie, Liefurung 3, Bern. (U.S.A. SIPRE Translation No. 14, 1954.)
Hallett, J., and Mason, B. J. (1958). The Influence of Temperature and Super saturation on the Habit of Ice Crystals Grown from the Vapour. Proc. R. Soc., A 247, 440-453.
Henson, E. W., and Longley, R. W. (1944). Meteorology Theoretical and Applied. Wiley,
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Hobbs, P. V . (1974). Ice Physics. Clarendon Press, Oxford.
Jost, W. (1960). Diffusion in Solids, Liquids, Gases. Academic Press, New York.
Keeler, C. M., and Weeks, W. F. (1967). Some Mechanical Properties of Alpine Snow, Mon­ tana 1964-66. C R R E L Report 227.
Keenan, J. H., Keyes, F. G., Hill, P. G., and Moore, J. G. (1978). Steam Tables. Wiley, New
York.

47
Kobayashi, T . (1957). Experimental Researches on the Snow Crystal Habit and Growth by
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LaChapeIIe, E. R. (1969). Field Guide to Snow Crystals. Univ. o f Washington Press.
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Luikov, A. V ., and Mikhailov, Yu. A. (1965). Theory of Energy and Mass Transfer. Oxford,
Pergamon Press, New York.
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Snow). Nauka, Moscow.

MONTANA STATE UNIVENSTTV t TWAWFS arch N378.L255
Studies on surface h o a r: form ation and